Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA II Lesson 17: Modeling with Polynomials—An Introduction Student Outcomes Students interpret and represent relationships between two types of quantities with polynomial functions. Lesson Notes In this lesson, students delve more deeply into modeling by writing polynomial equations that can be used to model a particular situation. Students are asked to interpret key features from a graph or table within a contextual situation (F-IF.B.4) and select the domain that corresponds to the appropriate graph or table (F-IF.B.5). Classwork Opening Exercise (8 minutes) Give students time to work independently on the Opening Exercise before discussing as a class. Opening Exercise In Lesson 16, we created an open-topped box by cutting congruent squares from each corner of a piece of construction paper. a. Express the dimensions of the box in terms of 𝒙. Length: 𝑳 = 𝟒𝟓. 𝟕 − 𝟐𝒙 Width: 𝑾 = 𝟑𝟎. 𝟓 − 𝟐𝒙 Height: 𝑯 = 𝒙 b. Write a formula for the volume of the box as a function of 𝒙. Give the answer in standard form. 𝑽(𝒙) = 𝒙(𝟒𝟓. 𝟕 − 𝟐𝒙)(𝟑𝟎. 𝟓 − 𝟐𝒙) 𝑽(𝒙) = 𝟒𝒙𝟑 − 𝟏𝟓𝟐. 𝟒𝒙𝟐 + 𝟏𝟑𝟗𝟑. 𝟖𝟓𝒙 How does this compare with the regression function found yesterday? Answers will vary. Compare each parameter in the function. Which function is more accurate? Why? The one found today is more accurate. The one found yesterday depended on measurements that may not have been exact. Lesson 17: Modeling with Polynomials—An Introduction This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE-1.3.0-07.2015 190 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA II Mathematical Modeling Exercises 1–13 (30 minutes) Allow students to work through the exercises in groups. Circulate the room to monitor students’ progress. Then, discuss results. Mathematical Modeling Exercises 1–13 The owners of Dizzy Lizzy’s, an amusement park, are studying the wait time at their most popular roller coaster. The table below shows the number of people standing in line for the roller coaster 𝒕 hours after Dizzy Lizzy’s opens. 𝒕 (hours) 𝑷 (people in line) 𝟎 𝟎 𝟏 𝟕𝟓 𝟐 𝟐𝟐𝟓 𝟒 𝟑𝟒𝟓 𝟕 𝟑𝟓𝟓 𝟖 𝟑𝟏𝟎 𝟏𝟎 𝟏𝟖𝟎 𝟏𝟐 𝟒𝟓 Jaylon made a scatterplot and decided that a cubic function should be used to model the data. His scatterplot and curve are shown below. Scaffolding: 1. Have early finishers find a quadratic model for comparison. Which model seems to be a better fit? Why? Answers will vary, but one model would be 𝑃(𝑡) = −10.134𝑡(𝑡 − 12). Students should see that the quadratic model does not seem to fit the data as well as the cubic model. They could confirm this using the regression feature of a graphing calculator. Do you agree that a cubic polynomial function is a good model for this data? Explain. Yes, the curve passes through most of the points and seems to fit the data. 2. What information would Dizzy Lizzy's be interested in learning about from this graph? How could they determine the answer? The company should be interested in the time when the line is the longest and how many people are in line at that time. To find this out, they can find a model that could be used to predict the number of people in line at any time during the day. They could then estimate the maximum point from the graph. 3. Estimate the time at which the line is the longest. Explain how you know. From the graph, the line is longest at 𝟓. 𝟓 hours because the relative maximum of the function occurs at 𝟓. 𝟓 hours. 4. Estimate the number of people in line at that time. Explain how you know. From the graph, there are roughly 𝟑𝟕𝟐 people in line when 𝒕 = 𝟓. 𝟓; that is the approximate relative maximum value of 𝑷. 5. Estimate the 𝒕-intercepts of the function used to model this data. The 𝒕-intercepts are roughly 𝟎, 𝟏𝟐. 𝟓, and 𝟑𝟑. 6. Use the 𝒕-intercepts to write a formula for the function of the number of people in line, 𝒇, after 𝒕 hours. 𝒇(𝒕) = 𝒄𝒕(𝒕 − 𝟏𝟐. 𝟓)(𝒕 − 𝟑𝟑), where 𝒄 is a constant that has not yet been determined. Lesson 17: Modeling with Polynomials—An Introduction This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE-1.3.0-07.2015 191 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA II Use the relative maximum to find the leading coefficient of 𝒇. Explain your reasoning. 7. Since we have estimated 𝒇(𝟓. 𝟓) = 𝟑𝟕𝟐, we can plug 𝟓. 𝟓 into the function above, and we find that 𝒇(𝟓. 𝟓) = 𝟏𝟎𝟓𝟖. 𝟕𝟓𝒄 = 𝟑𝟕𝟐. Solving for 𝒄, we find that 𝒄 ≈ 𝟎. 𝟑𝟓. The function that could model the data is then given by 𝒇(𝒕) = 𝟎. 𝟑𝟓𝒕(𝒕 − 𝟏𝟐. 𝟓)(𝒕 − 𝟑𝟑). What would be a reasonable domain for your function 𝒇? Why? 8. MP.3 A reasonable domain for 𝒇 would be 𝟎 ≤ 𝒙 ≤ 𝟏𝟐. 𝟓 because the opening of the park corresponds to 𝒕 = 𝟎, and after 𝟏𝟐. 𝟓 hours the park closes, so there are no people waiting in line. Use your function 𝒇 to calculate the number of people in line 𝟏𝟎 hours after the park opens. 9. The formula developed in Exercise 7 gives 𝒇(𝟏𝟎) = 𝟐𝟎𝟏. After the park has been open for 𝟏𝟎 hours, there will be 𝟐𝟎𝟏 people in line. 10. Comparing the value calculated above to the actual value in the table, is your function 𝒇 an accurate model for the data? Explain. The value of the function differs from the value from the table by about 𝟐𝟏 people, so it is not a perfect fit for the data, but it is otherwise very close. It appears to overestimate the number of people in line. 11. Use the regression feature of a graphing calculator to find a cubic function 𝒈 to model the data. The calculator gives 𝒈(𝒕) = 𝟎. 𝟒𝟑𝒕𝟑 − 𝟏𝟕. 𝟕𝟖𝒕𝟐 + 𝟏𝟓𝟔. 𝟔𝟑𝒕 − 𝟐𝟒. 𝟏𝟔. 12. Graph the function 𝒇 you found and the function 𝒈 produced by the graphing calculator. Use the graphing calculator to complete the table. Round your answers to the nearest integer. 𝒕 (hours) 𝑷 (people in line) 𝒇(𝒕) (your equation) 𝒈(𝒕) (regression eqn.) 𝟎 𝟎 𝟎 −𝟐𝟒 𝟏 𝟕𝟓 𝟏𝟐𝟗 𝟏𝟏𝟓 𝟐 𝟐𝟐𝟓 𝟐𝟐𝟖 𝟐𝟐𝟏 𝟒 𝟑𝟒𝟓 𝟑𝟒𝟓 𝟑𝟒𝟓 𝟕 𝟑𝟓𝟓 𝟑𝟓𝟎 𝟑𝟒𝟗 𝟖 𝟑𝟏𝟎 𝟑𝟏𝟓 𝟑𝟏𝟏 𝟏𝟎 𝟏𝟖𝟎 𝟐𝟎𝟏 𝟏𝟗𝟒 𝟏𝟐 𝟒𝟓 𝟒𝟒 𝟑𝟖 13. Based on the results from the table, which model was more accurate at 𝒕 = 𝟐 hours? 𝒕 = 𝟏𝟎 hours? At 𝒕 = 𝟐 hours, the function found by hand was more accurate. It was off by 𝟑 people whereas the calculator function was off by 𝟒 people. At 𝒕 = 𝟏𝟎 hours, the graphing calculator function was more accurate. It was off by 𝟏𝟒 people whereas the function found by hand was off by 𝟐𝟏 people. MP.2 Closing (2 minutes) What type of functions were used to model the data? Were they good models? What information did we use to find the function by hand? Cubic polynomial functions; yes, both functions were reasonably accurate. We used the 𝑥-intercepts and the relative maximum. Did we have to use the relative maximum specifically to find the leading coefficient? No, we could have chosen a different point on the curve. Lesson 17: Modeling with Polynomials—An Introduction This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE-1.3.0-07.2015 192 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M1 ALGEBRA II How did polynomials help us solve a real-world problem? We were able to model the data using a polynomial function. The function allows us to estimate the number of people in line at any time 𝑡 and also to estimate the time when the line is the longest and the maximum number of people are in line. Exit Ticket (5 minutes) Lesson 17: Modeling with Polynomials—An Introduction This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE-1.3.0-07.2015 193 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA II Name Date Lesson 17: Modeling with Polynomials—An Introduction Exit Ticket Jeannie wishes to construct a cylinder closed at both ends. The figure at right shows the graph of a cubic polynomial function, 𝑉, used to model the volume of the cylinder as a function of the radius if the cylinder is constructed using 150𝜋 cm3 of material. Use the graph to answer the questions below. Estimate values to the nearest half unit on the horizontal axis and to the nearest 50 units on the vertical axis. 1. What are the zeros of the function 𝑉? 2. What is the relative maximum value of 𝑉, and where does it occur? 3. The equation of this function is 𝑉(𝑟) = 𝑐(𝑟 3 − 72.25𝑟) for some real number 𝑐. Find the value of 𝑐 so that this formula fits the graph. 4. Use the graph to estimate the volume of the cylinder with 𝑟 = 2 cm. 5. Use your formula for 𝑉 to find the volume of the cylinder when 𝑟 = 2 cm. How close is the value from the formula to the value on the graph? Lesson 17: Modeling with Polynomials—An Introduction This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE-1.3.0-07.2015 194 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 17 M1 ALGEBRA II Exit Ticket Sample Solutions Jeannie wishes to construct a cylinder closed at both ends. The figure at right shows the graph of a cubic polynomial function, 𝑽, used to model the volume of the cylinder as a function of the radius if the cylinder is constructed using 𝟏𝟓𝟎𝝅 𝐜𝐦𝟑 of material. Use the graph to answer the questions below. Estimate values to the nearest half unit on the horizontal axis and to the nearest 𝟓𝟎 units on the vertical axis. 1. What are the zeros of the function 𝑽? Approximately 𝟎, −𝟖. 𝟓, and 𝟖. 𝟓 (Students might round up to −𝟗 and 𝟗.) 2. What is the relative maximum value of 𝑽 and where does it occur? The relative maximum value is 𝟖𝟎𝟎 𝐜𝐦𝟑 at = 𝟓 𝐜𝐦. 3. The equation of this function is 𝑽(𝒓) = 𝒄(𝒓𝟑 − 𝟕𝟐. 𝟐𝟓𝒓) for some real number 𝒄. Find the value of 𝒄 so that this formula fits the graph. Substituting 𝒓 = 𝟓 𝐜𝐦 and 𝑽(𝒓) = 𝟖𝟎𝟎 𝒄𝐦𝟑 and solving for 𝒄 gives 𝒄 ≈ −𝟑. 𝟒. 4. Use the graph to estimate the volume of the cylinder with 𝒓 = 𝟐 𝐜𝐦. Estimating from the graph, the volume of a cylinder of radius 𝟐 𝐜𝐦 is 𝟒𝟓𝟎 𝐜𝐦𝟑. 5. Use your formula for 𝑽 to find the volume of the cylinder when 𝒓 = 𝟐 𝐜𝐦. How close is the value from the formula to the value on the graph? Using the formula: 𝑽(𝟐) = −𝟑. 𝟒(𝟐𝟑 − 𝟕𝟐. 𝟐𝟓(𝟐)) = 𝟒𝟔𝟒. 𝟏. Therefore, the volume of the cylinder when 𝒓 = 𝟐 𝐜𝐦 is 𝟒𝟔𝟒. 𝟏 𝐜𝐦𝟑 . This value is close to the value of 𝟒𝟓𝟎 𝐜𝐦𝟑 found using the graph but not exact, particularly because we cannot read much detail from the graph. Problem Set Sample Solutions Problem 2 requires the use of a graphing calculator. If students do not have the means to complete this, the last two parts could be done in class. 1. Recall the math club fundraiser from the Problem Set of the previous lesson. The club members would like to find a function to model their data, so Kylie draws a curve through the data points as shown. a. What type of function does it appear that she has drawn? Degree 𝟑 polynomial (or cubic polynomial) Lesson 17: Modeling with Polynomials—An Introduction This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE-1.3.0-07.2015 195 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 17 NYS COMMON CORE MATHEMATICS CURRICULUM M1 ALGEBRA II b. The function that models the profit in terms of the number of t-shirts made has the form 𝑷(𝒙) = 𝒄(𝒙𝟑 − 𝟓𝟑𝒙𝟐 − 𝟐𝟑𝟔𝒙 + 𝟗𝟖𝟐𝟖). Use the vertical intercept labeled on the graph to find the value of the leading coefficient 𝒄. 𝒄 ≈ −𝟎. 𝟎𝟏𝟐𝟖𝟐, so 𝑷(𝒙) = −𝟎. 𝟎𝟏𝟐𝟖𝟐(𝒙𝟑 − 𝟓𝟑𝒙𝟐 − 𝟐𝟑𝟔𝒙 + 𝟗𝟖𝟐𝟖) c. From the graph, estimate the profit if the math club sells 𝟑𝟎 t-shirts. The profit is approximately $𝟐𝟓𝟎 if the club sells 𝟑𝟎 t-shirts. d. Use your function to estimate the profit if the math club sells 𝟑𝟎 t-shirts. 𝑷(𝟑𝟎) = 𝟐𝟑𝟎. 𝟏𝟒. The equation predicts a profit of $𝟐𝟑𝟎. 𝟏𝟒. e. Which estimate do you think is more reliable? Why? The estimate from the graph is probably more reliable because the equation required estimating the 𝒙-intercepts. If these estimates were off, it could have affected the equation. 2. A box is to be constructed so that it has a square base and no top. a. Draw and label the sides of the box. Label the sides of the base as 𝒙 and the height of the box as 𝒉. b. The surface area is 𝟏𝟎𝟖 𝐜𝐦𝟐. Write a formula for the surface area 𝑺, and then solve for 𝒉. 𝑺 = 𝒙𝟐 + 𝟒𝒙𝒉 = 𝟏𝟎𝟖 𝒉= c. 𝟏𝟎𝟖 − 𝒙𝟐 𝟒𝒙 Write a formula for the function of the volume of the box in terms of 𝒙. 𝟏𝟎𝟖 − 𝒙𝟐 𝟏𝟎𝟖𝒙𝟐 − 𝒙𝟒 𝟏𝟎𝟖𝒙 − 𝒙𝟑 𝑽(𝒙) = 𝒙𝟐 𝒉 = 𝒙𝟐 ( = )= 𝟒𝒙 𝟒𝒙 𝟒 d. Use a graphing utility to find the maximum volume of the box. 𝟏𝟎𝟖 𝐜𝐦𝟑 e. What dimensions should the box be in order to maximize its volume? 𝟔 𝐜𝐦 × 𝟔 𝐜𝐦 × 𝟑 𝐜𝐦 Lesson 17: Modeling with Polynomials—An Introduction This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG II-M1-TE-1.3.0-07.2015 196 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.